# Circular Tank Design problem

Design a circular tank, closed at both ends, with a volume of 200 cubic metres .The cost is proportional to the surface area of material, which is priced at \$400per sqaure metre. The tank is contained within a shed with a sloping roof, thus the height of the tank h is limited by where d is the tank diameter. Formulate the minimum cost problem and solve the design problem.

Mathematical Formulation:

Minimize: Subjected to: ```clc; function cost=Tankprob(x) r = x(1);h = x(2); cost = (2*%pi*r*h+2*%pi*r^2)*400; endfunction function [ceq, c]=Nonlinearcon(x) r = x(1);h = x(2); c = []; ceq = 200-%pi*r^2*h; endfunction function y=Gradobj(x) y= [800*%pi*x(2) + 1600*%pi*x(1),800*%pi*x(1)]; endfunction mprintf('\nDesign a circular tank, closed at both ends, with a volume of 200 m3 with minimum cost.\n The tank is contained within a shed with a sloping roof,thus the height of the tank h is limited') mprintf('\nCost of material is: %f',400); mprintf('The design variables are radius and height of the tank') A = [1 1];b = 12; x0 = input('Enter initial guess as vector:'); if (sum(x0<=0) | (length(x0)~=2)) x0 = [1 2]; mprintf('Incorrect initial guess...\n changing initial guess to r = %d and h = %d',x0(1),x0(2)); end lb = [0 0]; input('press enter to continue') options=list("MaxIter",1000,"GradObj", Gradobj); [xopt,fopt,exitflag,output1] = fmincon(Tankprob,x0,A,b,[],[],lb,[],Nonlinearcon,options); [xopt1,fopt1,exitflag1,output2] = fmincon(Tankprob,x0,A,b,[],[],lb,[],Nonlinearcon); clc if exitflag == 0 then mprintf('Optimal Solution Found'); mprintf('\nThe optimal radius and height of the tank are %f m and %f m',xopt(1),xopt(2)); mprintf('\nThe volume of the tank is %f m^3',%pi*xopt(1)^2*xopt(2)); mprintf('\nThe total surface area and cost of the tank are %f m^2 and %f \$',fopt/400,fopt) mprintf('\nTime taken to solve the problem with gradient information is %f s and without gradient information is %f s',output1.Cpu_Time,output2.Cpu_Time); elseif exitflag == 1 then mprintf('Maximum Number of Iterations Exceeded. Output may not be optimal'); input('press enter to view results'); printf('\nThe optimal radius and height of the tank are %f m and %f m',xopt(1),xopt(2)); mprintf('\nThe volume of the tank is %f m^3',%pi*xopt(1)^2*xopt(2)); mprintf('\nThe total surface area and cost of the tank are %f m^2 and %f \$',fopt/400,fopt) else disp("Error encountered") end```

Expected Output:

 ```Optimal Solution Found The optimal radiusand height of the tank are 3.169203 m and 6.338406 m The volume of the tank is 200.000000 m^3 The total surface areaand cost of the tank are 189.322054 m^2 and 75728.821670 \$ Time taken to solve the problem with gradient information is 0.181715 s and without gradient information is 0.194686 s```